×

Which properties of a random sequence are dynamically sensitive? (English) Zbl 1021.60055

A lot of classical results in probability theory, for instance the strong law of large numbers, the law of iterated logarithm, and so on, concern almost-sure properties of sequences \(\{X_n\}\) of i.i.d. random variables.
Consider a sequence of i.i.d. random variables \(\{X_n\}\) with state space \(S\) and common law \(\nu\), where each variable is replaced independently by an independent variable with the same law according to a Poisson clock, i.e., for each \(n\in\mathbb{N},\) let \(X(t):=\{X_n(t)\},\) \(t\geq 0\), be an independent process which at rate 1 replaces its current value by an independent sample from \(\nu.\) The distribution of \(X(t)\) is \(\mu=\nu^{\mathbb{N}}\) for every \(t\geq 0,\) thus any Borel set \(A\subset S^{\mathbb{N}}\) such that \(\mu(A)=1\) satisfies for all \(t\geq 0,\) \(P\{X(t)\in A\}=1\) and \(P\{X(t)\in A\text{ for Lebesgue-a.e. }t\}=1.\) The almost-sure properties of \(\{X_n\}\), i.e., the subsets \(A\) of \(S^{\mathbb{N}}\) with \(\mu(A)=1\) are classified by the authors as (dynamically) stable or sensitive according to whether or not the property: for all \(t\geq 0,\) \(P\{X(t)\in A\text{ for Lebesgue-a.e. }t\}=1\) can be strengthened to \(P\{X(t)\in A\text{ for all }t\}=1\).
The authors prove that the law of large numbers and the law of iterated logarithm are dynamically stable while run tests are dynamically sensitive. They obtain multi-fractal analysis of exceptional times for run lengths and for prediction.
For the simple random walk in \(\mathbb{Z}^d\) the authors prove that the transience is dynamically stable for \(d\geq 5,\) and dynamically sensitive for \(d=3,4.\) Moreover, for \(d=3,4\) the nonempty random set of exceptional times \(t\) where the random walk is recurrent has Hausdorff dimension \((4-d)/ 2\) a.s.

MSC:

60J25 Continuous-time Markov processes on general state spaces
28A80 Fractals
60F15 Strong limit theorems
28A78 Hausdorff and packing measures
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] ADELMAN, O., BURDZY, K. and PEMANTLE, R. (1998). Sets avoided by Brownian motion. Ann. Probab. 26 429-464. · Zbl 0934.60016
[2] ALON, N. and SPENCER, J. (1992). The Probabilistic Method. Wiley, New York.
[3] BENJAMINI, I., KALAI, G. and SCHRAMM, O. (1999). Noise sensitivity of Boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math. 90 5-43. · Zbl 0986.60002
[4] BENJAMINI, I. and PERES, Y. (1994). Tree-indexed random walks on groups and first passage percolation. Probab. Theory Related Fields 98 91-112. · Zbl 0794.60068
[5] BENJAMINI, I. and SCHRAMM, O. (1998). Exceptional planes of percolation. Probab. Theory Related Fields 111 551-564. · Zbl 0910.60076
[6] DEMBO, A., PERES, Y., ROSEN, J. and ZEITOUNI, O. (2000). Thin points for Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 36 749-774. · Zbl 0977.60073
[7] DURRETT, R. (1996). Probability: Theory and Examples, 2nd ed. Wadsworth, Belmont, CA. · Zbl 0709.60002
[8] ERD OS, P. and RÉVÉSZ, P. (1977). On the length of the longest head-run. Colloq. Math. Soc. János Boly ai 16 219-228.
[9] FUKUSHIMA, M. (1984). Basic properties of Brownian motion and a capacity on the Wiener space. J. Math. Soc. Japan 36 161-176. · Zbl 0535.60068
[10] HÄGGSTRÖM, O. (1998). Dy namical percolation: early results and open problems. In Microsurvey s in Discrete Probability (D. Aldous and J. Propp, eds.) 59-74. Amer. Math. Soc., Providence, RI. · Zbl 0906.60081
[11] BENJAMINI, HÄGGSTRÖM, PERES AND STEIF
[12] HÄGGSTRÖM, O., PERES, Y. and STEIF, J. E. (1997). Dy namical percolation. Ann. Inst. H. Poincaré Probab. Statist. 33 497-528. · Zbl 0894.60098
[13] HEBISCH, W. and SALOFF-COSTE, L. (1993). Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21 673-709. · Zbl 0776.60086
[14] KAHANE, J. P. (1985). Some Random Series of Functions, 2nd ed. Cambridge Univ. Press. · Zbl 0571.60002
[15] KHOSHNEVISAN, D., PERES, Y. and XIAO, Y. (2000). Limsup random fractals. Electron. J. Probab. 5. · Zbl 0949.60025
[16] KOLMOGOROV, A. N. and USPENSKII, V. A. (1987). Algorithms and randomness. Theory Probab. Appl. 32 389-412. · Zbl 0648.60005
[17] KÔNO, N. (1984). 4-dimensional Brownian motion is recurrent with positive capacity. Proc. Japan Acad. Ser. A Math. Sci. 60 57-59. · Zbl 0559.60063
[18] KUELBS, J. (1976). A strong convergence theorem for Banach space valued random variables. Ann. Probab. 4 744-771. · Zbl 0365.60034
[19] LI, M. and VITÁNy I, P. (1993). An Introduction to Kolmogorov Complexity and Its Applications. Springer, New York.
[20] MATTILA, P. (1995). Geometry of Sets and Measures in Euclidean Spaces. Cambridge Univ. Press. · Zbl 0819.28004
[21] PARTHASARATHY, K. R. (1967). Probability Measures on Metric Spaces. Academic Press, New York. · Zbl 0153.19101
[22] PERES, Y. (1996). Remarks on intersection-equivalence and capacity-equivalence. Ann. Inst. H. Poincaré Phy s. Théor. 64 339-347. · Zbl 0854.60077
[23] PERES, Y. and STEIF, J. E. (1998). The number of infinite clusters in dy namical percolation. Probab. Theory Related Fields 111 141-165. · Zbl 0906.60069
[24] SPITZER, F. (1976). Principles of Random Walk, 2nd ed. Springer, New York. · Zbl 0359.60003
[25] BERKELEY, CALIFORNIA 94720 AND INSTITUTE OF MATHEMATICS HEBREW UNIVERSITY JERUSALEM ISRAEL E-MAIL: peres@stat.berkeley.edu www.stat.berkeley.edu/ peres J. E. STEIF SCHOOL OF MATHEMATICS GEORGIA INSTITUTE OF TECHNOLOGY URL:
[26] ATLANTA, GEORGIA 30332-1060 AND CHALMERS UNIVERSITY OF TECHNOLOGY GOTHENBURG SWEDEN E-MAIL: steif@math.gatech.edu www.math.chalmers.se/ steif URL:
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.